# Questions tagged [probability]

A probability expresses quantitatively how likely an event is to occur. We often encounter probabilities as conditional probabilities which express how likely an event is to occur in light of certain (given) information.

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### Distribution of hitting time of the integrated CIR process

If an increasing process $X_t$ has a known Laplace transform $\mathbb{E} e^{-s X_t} = m_t(s)$, define its hitting time $\tau$ to some level $B$ to be $$\tau = \inf\{ u > 0 : X_u \geq B \}.$$ Can ...
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### 2-state HMM / ARMA process?

I have issues with this problem: Let $\{X_t, t\in \Bbb N\}$ be a 2-state stationary Markov chain, with transition $M$ (and $M(1,2)\neq 0 \neq M(2,1)$), let $\{W_t, t\in \Bbb N\}$ be a strong Gaussian ...
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### Transition densities in the Heston model

Knowing the Characteristic function $\Phi_{T,t} = \mathbb{E} [ e^{i u S_T} | S_t, V_t]$ (or equivalently, the Laplace transform) of an affine process, it's possible to know the distribution of the ...
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### Fitting Student t-distributions to log-returns

It seems that some tail-risk centric groups are bent on using Paretian and t-distributions to account for tail risk when fitting log-returns. It has been observed, however, that with and without ...
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### How do I calculate the present value of a credit default swap?

I am paid 20 million every time a bond drops to a new low over a 120 month period. I need to know how to find the present value of such an arrangement if there is a continuously compound interest of 5 ...
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258 views

### negative transition probabilities in the heston model

I've been trying to implement a bivariate tree for pricing american options with the heston model in R using the paper of Beliaeva and Nawalkha (http://papers.ssrn.com/sol3/papers.cfm?abstract_id=...
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698 views

### Monty Hall Model

Given a fixed time period,say 3 days, the stock/market can go up,down or stay sideways. A hedge fund can long, short or use rangebound(options strategy) to bet for that 3 days closing level. Hedge ...
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### GBM probability of hitting non constant barrier

I know there is a formula for probability of hitting a constant barrier for GBM/BM (See page 651 in Martinagle Methods in Financial Modelling). Is there a formula for non-constant barrier? The ...
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### Law of a geometric brownian motion first hitting time (formula dont match Monte Carlo Simulation)

I posted this question before on MSE I need to use it in a small step in the middle of a simulation and I think I'm not getting correct results to this probabilities and so for my all ...
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240 views

### default probability

Suppose the hazard rate is $\lambda$ the default probability density function follow exponential $f(t) = \lambda e^{-\lambda t}$ and cumulative probability function is $F(t) = 1 - e^{-\lambda t}$ ...
1answer
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### Unconditional Expectation vs. Conditional Expectation at time $0$

In most mathematical finance books I have read (all of them actually), the expectation, with respect to the sigma algebra at time $0$, $\mathcal F_0$, is considered the same as the unconditional ...
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401 views

### Detecting butterfly spread arbitrage for American options through European option prices

It's easy to demonstrate that if European option prices are concave with strike, then an arbitrage exists. For example, the risk-neutral probability density is the second derivative of European put ...
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### Laplace Exponent of a Jump-Diffusion Process

I'm currently reading a paper (https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2543702) which uses the following process to describe the dynamics of a firm's asset value: V_t = ...
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### Portfolio diversification on default risk

A portfolio of 13 different companies have loans. Company $i$ default on their loan with probability $p_i$ and survive with prob $q_i=1-p_i$. Let $Y_i=1$ denote default. Question: How could I get to a ...
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### interview question : replication strategy of a betting game

Here is a question I found in a book I am not able to finish. Your help will be much appreciated! I also included where I have been so far. Q: Team A plays team B in a series of 7 games, whoever wins ...
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177 views

### Beta distribution - Holding period

Let's say I have a risk factor that is defined between [0,1], such as recovery rates. Assuming I have daily data, I can estimate the "daily VaR", i.e. the tails over 1 day period, since the data is ...
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231 views

### Probability Density of Returns of Bonus Certificates

Could anyone please help me with the following? I need to generate a histogram (resp. probability density) of returns of a bonus-certificate. A bonus-certificate can be replicated by an underlying ...
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247 views

### Modeling asset performance to Bitcoin revenue

I'm attempting to model asset performance to Bitcoin revenue, which is a driving force in the Bitcoin community. Question Is there any model, or research being done that tracks "hashes per second" (...
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### Introducting a new probability measure

I'm trying to understand what means : $$\frac {d \mathbb {\tilde{P}} }{d \mathbb P } \bigg\rvert_{\mathcal F_t }$$where $\mathcal F_t$ is a filtration I guess (not explicitely mentionned). they ...
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### How are Risk indices linked to Physical Trading returns?

Ref to my previous question here: Physical trading spot transaction analysis-Quantified I have been able to narrow down my aim to defining a physical trading strategy P&L. My question is, how ...
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196 views

### How to compute SABR's probability density function

I am trying to compute the probability density function of the forward rate implied by the SABR formula approximation in order to see how the density implied by the approximation has negative ...
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### Computing the PDF of the sum of N moves of an empirical PDF for USDJPY 1-minute moves

Per-minute tick data for USDJPY is available here. Suppose we download this file to usdjpy.txt and then save it into a Numpy array in Python 3 as follows: ...
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### A priori selection of acceptable backtesting errors (type I and II)

Is it possible to a priori select an acceptable values of type I and II errors in backtesting (f.e. in case of the unconditional coverage test)? Type I error is directly connected to the significance ...
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### Density of a portfolio's returns is the weighted average of asset distributions?

The expected return of a portfolio can be formulated as a weighted average of the constituent assets' returns: $$r_p = w_1 r_1 + w_2 r_2 + \dots + w_N r_N + \epsilon$$ Does it also follow that the ...
1answer
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### Bayesian analysis in R for low default portfolios

I want to apply the knowledge of this paper (Bayesian estimation of probabilities of default for low default portfolios, by Dirk Tasche) in R, but I can't find the right bayesian package and functions ...
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### What is the meaning of this notation, D lag t?

I'm reading the book Financial Markets Under the Microscope to study market microstructure. There is a notation that I could not understand. What is the meaning of D here? It is not used in the text ...
1answer
82 views

### Convert option inputs to standard Brownian motion

I want to know the probability that the strike price of an option is touched. My input values are: P = price S = strike v = vol t = time to expiration According ...
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### Geometric brownian motion and probabilities

A stock's price movement is described by the equations $dS_t=0.02S_tdt+0.25S_tdW_t$ and $S_0=100$. An investor buys a call option on said stock with a strike price $K=95$ which expires in $T=2$ years. ...
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### Specify user-defined distribution for multivariate distribution in copula R package

For the copula R package, the function Mvdc allows the margins to be user-defined. ...
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59 views

### Probability and random walk

Let's says i have 10 years of daily prices on a stock ABC. I do some analysis and I realise that, for example, if the stock increases 5 days in a row (close > open), 75% of the time, the 6th day will ...
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### Sample conditional multivariate random variable?

There's multivariate random variable, future prices of assets, 5 years from now: $$X = [Gold, Silver, SP500]$$ There's historical prices for $X$ available for last 50 years. It's possible to fit ...